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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000125
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St000125: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0
[.,[.,.]]
=> 0
[[.,.],.]
=> 0
[.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> 0
[[.,.],[.,.]]
=> 0
[[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> 0
[.,[[.,.],[.,.]]]
=> 0
[.,[[.,[.,.]],.]]
=> 0
[.,[[[.,.],.],.]]
=> 1
[[.,.],[.,[.,.]]]
=> 0
[[.,.],[[.,.],.]]
=> 0
[[.,[.,.]],[.,.]]
=> 0
[[[.,.],.],[.,.]]
=> 0
[[.,[.,[.,.]]],.]
=> 0
[[.,[[.,.],.]],.]
=> 0
[[[.,.],[.,.]],.]
=> 0
[[[.,[.,.]],.],.]
=> 0
[[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> 0
[.,[.,[[[.,.],.],.]]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> 1
[.,[[[[.,.],.],.],.]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> 0
[[[.,.],.],[[.,.],.]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> 0
[[[[.,.],.],.],[.,.]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[.,[[[.,.],.],.]]}}} in a binary tree.
[[oeis:A005773]] counts binary trees avoiding this pattern.
Matching statistic: St000871
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => ? = 0
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [7,4,5,6,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [6,4,5,7,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [5,3,2,1,4,6,7] => ? = 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 0
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [7,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [7,4,5,3,6,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [7,5,3,4,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [7,4,3,5,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [6,4,5,3,7,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [5,3,2,1,4,6,7] => ? = 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [6,5,3,4,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [6,4,3,5,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 0
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 0
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => ? = 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 0
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [7,4,5,6,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [6,4,5,7,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [5,4,6,7,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 1
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [6,4,3,2,1,5,7] => ? = 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [7,5,6,3,2,4,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => ? = 1
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [7,6,5,2,3,4,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [6,5,2,1,3,4,7] => ? = 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> [7,5,6,2,3,4,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 1
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [6,5,7,2,3,4,1] => [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [6,5,3,2,1,4,7] => ? = 2
Description
The number of very big ascents of a permutation.
A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$.
For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
Matching statistic: St001330
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,.]]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[.,.],.]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[[[.,.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[.,.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[[[.,.],[.,.]],[.,[.,.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[[[.,[.,.]],.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[.,.],[.,[.,.]]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[[[.,[.,[.,.]]],.],[.,.]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[[.,.],[.,.]],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[[.,[.,.]],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[.,[.,.]],[.,[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000660
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1 + 1
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[.,[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 1 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[[.,.],[.,[.,[[.,.],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[[.,.],[.,[[.,[.,.]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[[.,.],[.,[[[.,.],.],.]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[[.,.],[[.,.],[[.,.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[[.,.],[[[.,.],.],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[[.,.],[[.,[.,[.,.]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[[.,.],[[.,[[.,.],.]],.]]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[[.,.],[[[.,.],[.,.]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 1
[[.,.],[[[.,[.,.]],.],.]]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[[.,[.,.]],[.,[[.,.],.]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 0 + 1
[[.,[.,.]],[[.,.],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[[.,[.,.]],[[.,[.,.]],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001719
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [1] => [1] => ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[.,.],.]
=> [1,2] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,5,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [4,2,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [5,4,3,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,4,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [5,4,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [5,4,2,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [5,4,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [5,3,4,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [5,3,4,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [5,3,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [6,3,4,5,2,1] => [5,2,3,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [5,3,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [5,3,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [5,3,4,6,2,1] => [5,2,3,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [5,2,1,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> [3,4,5,6,2,1] => [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 1 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => [4,5,3,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [5,6,4,2,3,1] => [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [4,5,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
=> ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [5,4,6,2,3,1] => [4,5,2,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,11),(11,10)],12)
=> ? = 0 + 1
[[[.,.],[.,.]],[.,[.,.]]]
=> [6,5,3,1,2,4] => [5,3,6,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,.],[.,.]],[[.,.],.]]
=> [5,6,3,1,2,4] => [5,3,6,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],[.,.]]
=> [6,4,3,1,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],[.,.]]
=> [6,3,4,1,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => [3,5,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => [4,5,2,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],[.,.]]
=> [6,3,1,2,4,5] => [4,2,5,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,[.,.]],[.,[.,.]]],.]
=> [5,4,2,1,3,6] => [3,5,2,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,[.,.]],[[.,.],.]],.]
=> [4,5,2,1,3,6] => [3,5,1,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],[.,.]],.]
=> [5,3,2,1,4,6] => [3,2,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[.,[[.,.],.]],[.,.]],.]
=> [5,2,3,1,4,6] => [2,3,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => [4,2,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
[[[[.,[.,.]],.],[.,.]],.]
=> [5,2,1,3,4,6] => [2,4,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[[.,[.,.]],[.,.]],.],.]
=> [4,2,1,3,5,6] => [2,4,1,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
[[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => [5,3,6,2,7,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
[[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => [3,5,2,6,1,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 1 = 0 + 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001720
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [1] => [1] => ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,.]]
=> [2,1] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[.,.],.]
=> [1,2] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[.,[.,[.,.]]]
=> [3,2,1] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,.],.],.]
=> [1,2,3] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 0 + 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2 = 0 + 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2 = 0 + 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 0 + 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 0 + 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 0 + 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 0 + 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,5,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 2
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 0 + 2
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 2
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 2
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? = 0 + 2
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 2
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [4,2,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 2
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 2
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 2
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [5,4,3,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 0 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [5,4,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [5,4,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 0 + 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [5,4,2,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ? = 0 + 2
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [5,4,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? = 1 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [5,3,4,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
=> ? = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [5,3,4,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
=> ? = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [5,3,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 0 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> [6,3,4,5,2,1] => [5,2,3,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
=> ? = 1 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [5,3,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ? = 0 + 2
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [5,3,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 0 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> [5,3,4,6,2,1] => [5,2,3,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1 + 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [5,2,1,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 1 + 2
[.,[.,[[[[.,.],.],.],.]]]
=> [3,4,5,6,2,1] => [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 1 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => [4,5,3,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
=> [5,6,4,2,3,1] => [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
=> ? = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [4,5,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
=> ? = 0 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [5,4,6,2,3,1] => [4,5,2,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,11),(11,10)],12)
=> ? = 0 + 2
[[[.,.],[.,.]],[.,[.,.]]]
=> [6,5,3,1,2,4] => [5,3,6,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,.],[.,.]],[[.,.],.]]
=> [5,6,3,1,2,4] => [5,3,6,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,.],[.,[.,.]]],[.,.]]
=> [6,4,3,1,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,.],[[.,.],.]],[.,.]]
=> [6,3,4,1,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => [3,5,2,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
[[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => [4,5,2,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[[.,.],[.,.]],.],[.,.]]
=> [6,3,1,2,4,5] => [4,2,5,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,[.,.]],[.,[.,.]]],.]
=> [5,4,2,1,3,6] => [3,5,2,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,[.,.]],[[.,.],.]],.]
=> [4,5,2,1,3,6] => [3,5,1,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,[.,[.,.]]],[.,.]],.]
=> [5,3,2,1,4,6] => [3,2,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[.,[[.,.],.]],[.,.]],.]
=> [5,2,3,1,4,6] => [2,3,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => [4,2,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
[[[[.,[.,.]],.],[.,.]],.]
=> [5,2,1,3,4,6] => [2,4,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[[.,[.,.]],[.,.]],.],.]
=> [4,2,1,3,5,6] => [2,4,1,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
[[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => [5,3,6,2,7,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
[[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => [3,5,2,6,1,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> 2 = 0 + 2
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001882
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [3,2,1] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [4,2,1,3] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [6,1,2,3,5,4] => [6,1,2,3,5,4] => ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => [6,1,2,4,3,5] => [6,1,2,4,3,5] => ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [6,1,2,5,4,3] => [6,1,2,5,4,3] => ? = 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [6,1,2,4,5,3] => [6,1,2,4,5,3] => ? = 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => [6,1,3,2,4,5] => [6,1,3,2,4,5] => ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => [6,1,3,2,5,4] => [6,1,3,2,5,4] => ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [4,3,6,5,2,1] => [6,1,4,3,2,5] => [6,1,4,3,2,5] => ? = 0
[.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => [6,1,3,4,2,5] => [6,1,3,4,2,5] => ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [6,1,5,3,4,2] => [6,1,5,3,4,2] => ? = 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [6,1,5,4,3,2] => [6,1,5,4,3,2] => ? = 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [3,5,4,6,2,1] => [6,1,3,5,4,2] => [6,1,3,5,4,2] => ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [6,1,4,3,5,2] => [6,1,4,3,5,2] => ? = 1
[.,[.,[[[[.,.],.],.],.]]]
=> [3,4,5,6,2,1] => [6,1,3,4,5,2] => [6,1,3,4,5,2] => ? = 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [2,6,5,4,3,1] => [6,2,1,3,4,5] => [6,2,1,3,4,5] => ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [2,5,6,4,3,1] => [6,2,1,3,5,4] => [6,2,1,3,5,4] => ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [2,4,6,5,3,1] => [6,2,1,4,3,5] => [6,2,1,4,3,5] => ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [2,5,4,6,3,1] => [6,2,1,5,4,3] => [6,2,1,5,4,3] => ? = 0
[.,[[.,.],[[[.,.],.],.]]]
=> [2,4,5,6,3,1] => [6,2,1,4,5,3] => [6,2,1,4,5,3] => ? = 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [3,2,6,5,4,1] => [6,3,2,1,4,5] => [6,3,2,1,4,5] => ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [3,2,5,6,4,1] => [6,3,2,1,5,4] => [6,3,2,1,5,4] => ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [2,3,6,5,4,1] => [6,2,3,1,4,5] => [6,2,3,1,4,5] => ? = 1
Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St000181
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [1,0]
=> [1,0]
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001490
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [.,.]
=> [1,0]
=> [[1],[]]
=> 1 = 0 + 1
[.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> 1 = 0 + 1
[[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 0 + 1
[.,[[[.,.],.],.]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 1 + 1
[[.,.],[.,[.,.]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 0 + 1
[[[[.,.],[.,.]],.],.]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? = 0 + 1
[[[[.,[.,.]],.],.],.]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? = 0 + 1
[[[[[.,.],.],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5],[3]]
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [[.,[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [[.,[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[5,3],[1]]
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [[.,[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [[.,[[.,[.,.]],[.,.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [[.,[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [[.,[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [[.,[[[.,.],[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [[.,[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[5,5],[1]]
=> ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> [[.,[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 1 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [[[.,.],[.,[[.,.],.]]],.]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [[[.,.],[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1]]
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> [[[.,.],[[[.,.],.],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [[[.,[.,.]],[[.,.],.]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[3,3,2,2],[1]]
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [[[[.,.],.],[.,[.,.]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> ? = 1 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St001890
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [1,0]
=> [1,0]
=> ([],1)
=> ? = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0 + 1
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0 + 1
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 0 + 1
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001811The Castelnuovo-Mumford regularity of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.
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